Upside Down Primes
 

The digits 0, 1, 2, 5, 6, 8, and 9 have a kind of collective symmetry
upon turning a decimal number 180 degrees (clockwise), with 0, 1, and 8
transforming to themselves, 2 and 5 also (using some imagination),
and 6 and 9 transforming into each other.

If a number and its upside down transform are both prime (and different),
like 19 and 61, or 109 and 601, these numbers are "partners".

Find as large a pair of partners as possible that contain all 7 of these
digits at least once. (I include the last condition because it might be
too easy to find a rep-unit or a form 10^n + 9).

Starting off with a pair of "small" ones:

1<97>25689<63>1 and 1<63>68952<97>1 are a pair of 167 digit partners. The part in <> is the number of zeroes.

EDIT:- 305 digit solutions: 1<271>25689<27>1 and 1<27>68952<271>1

EDIT2:- 404 digits: 1<344>25689<53>1 and 1<53>68952<344>1

Slight improvement with a different pattern set.

426 digits: 1<386>98652<33>1 and 1<33>25986<386>1

two better results

595 digits: 1<276>98652<312>1 and 1<312>25986<276>1
596 digits: 1<401>98652<188>1 and 1<188>25986<401>1

Excellent results, a few (hundred) digits longer than I anticipated.

May I ask: since so far your prime finds all have the same basic pattern
(start and end in 1, many 0's, and one splash of the other five digits
somewhere in the middle), is your algorithm to generate increasingly
large numbers with these patterns and then test for dual-primality? If so,
it should be only a little harder to find a pair that "look" more random,
say by not having any repeated adjacent digits. All those zeros make
the generating pattern evident.

Not that there's anything wrong with that. :smile:

[QUOTE=davar55;235707]May I ask: since so far your prime finds all have the same basic pattern
(start and end in 1, many 0's, and one splash of the other five digits
somewhere in the middle), is your algorithm to generate increasingly
large numbers with these patterns and then test for dual-primality? If so,
it should be only a little harder to find a pair that "look" more random,
say by not having any repeated adjacent digits. All those zeros make
the generating pattern evident. [/QUOTE]

Indeed. I chose the approach for it's automatability, and brain-dead-ness:smile: -- takes all of two lines of pari code. Not sure I want to go to the randomness route -- I would have to think about how to automate that.

And the first titanic partners:

1110 digits: 1<878>98652<225>1 and 1<225>25986<878>1

Just because I'm not as familiar with the tools as I should be,
are these proven primes or probable primes? How do you check
a basically random 1110 digit number for proven primality?
If it's just a pari feature, do you know how it does it?

Pari's not good at proving primes of that size.

1420 digits: 1<431>98652<982>1 and 1<982>25986<431>1

Searching is only prp tests. All the smaller ones were proven using PARI. For the largest one, I used PARI to prove one of them quickly (the one with more trailing zeroes). Currently running PRIMO on the other.

EDIT:- The other one was just proved by pari in 15 minutes using APR-CL. On to the 1420 digits.

Primo has also certified the 1110 digit prime. Pari has proven the 1420 digit prime in 40 mins. Primo is currently running on it.